「～～でもない」, in that particular context, means "not worth ~~". Thus, the second half of the sentence in question would mean:
"I don't think it's something worth making such a fuss over"
Or, more literally, you could change the "something" to "a story" since the word used in the original is 「話」.
What 「でもない」 means depends heavily on the context as Japanese is an extremely contextual language. Here and in many other situations, it means "something is rather trivial".
Moving on to how to use 「でもない」... Permit me to just give an example for each of the definitions you have copied from the dictionary.
1) 1.(exp) neither... nor... (with the implication that there are other "nor..." items)
For this usage, you must use 「でも」 at least twice as in:
「これはラーメンでもうどんでもない。『ラードン』だ。」 (I invented a noodle dish.)
2) denying explicitly one thing and also denying implicitly other things as well
"Not only did I not work, I didn't really do anything else this past year." (Not a direct translation, but you know what I mean.)
3) not like (you); not (your) usual self
「そんなことをするなんて、お前でもないよ。」("It isn't like you to do such a thing.")
I myself feel this usage is kind of old-fashioned. We would use 「～～らしくない」 nowadays. (Perhaps I'm too young is all.)